

Here, \(Q:=\Omega \times (0,T)\) , where \(\Omega \subset \mathbb {R}^d\)  with \(d= 2,3\)  denotes a bounded, smooth domain with boundary \(\Gamma \)  and \(T>0\)  is a given final time.
In (), \(\rho \)  is the constant density of the mixture, \({v}\)  is the velocity of the mixture, \(p\)  is the pressure, \(\phi \)  is the difference of the fluid concentrations, \(\mu \)  is the chemical potential. In addition, \(\nu \)  is the concentration depending viscosity, \( is the symmetric gradient operator, \)\( is a positive parameter related to the thickness of the interface, \) m\( is the (bulk) mobility function and \) F\( is a double-well potential.\) The fundamental assumption in deriving the model H is that the density of the mixture is constant, meaning that the two fluids have the same constant density \(\rho \)  (matched densities). In the past two decades, several works investigated suitable Navier–Stokes–Cahn–Hilliard generalizations of the model H aiming to describe both incompressible mixtures with unmatched densities and compressible two-phase flows. Without claiming of completeness, we refer the reader to [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}.
Among them is the thermodynamically consistent model for viscous incompressible mixtures with unmatched densities proposed by Abels, Garcke and Grün (AGG model) in the seminal work [1]}, which reads as follows:
\(&\partial _{t}\big (\rho (\phi ) {v}\big )+ \textnormal {div}\big ( {v}\otimes (\rho (\phi ){v}+ J) \big )- \textnormal {div}\big ( 2\nu (\phi ) \, \big )+ \nabla p= - \varepsilon \, \textnormal {div}\big (\nabla \phi \otimes \nabla \phi \big )&&\quad \text{in $Q$},\\[1ex]&\textnormal {div}\, {v}= 0&&\quad \text{in $Q$},\\[1ex]&\rho (\phi )= \tilde{\rho }_1 \frac{1+\phi }{2}+ \tilde{\rho }_2 \frac{1-\phi }{2},\quad J= - \tfrac{1}{2} (\tilde{\rho }_2 - \tilde{\rho }_1)m_\Omega (\phi ) \nabla \mu &&\quad \text{in $Q$},\\[1ex]&\partial _{t}\phi + \textnormal {div}(\phi {v})= \textnormal {div}\big ( m_\Omega (\phi ) \nabla \mu \big )&&\quad \text{in $Q$},\\[1ex]&\mu = -\varepsilon \Delta \phi + \varepsilon ^{-1} F^{\prime }(\phi )&&\quad \text{in $Q$}.\) 
